Conventional magnetic resonance imaging experiments are designed to give optimal resolution on macroscopic length scales characteristic of human anatomical features and their pathologies (i.e. 0.0001 to 0.1 meters). Because of the obvious advantages of MRI as a rapid, non-invasive diagnostic technique there is a great deal of interest in pushing the resolution of MRI to length scales typical of cellular structures. Physical phenomena such as diffusion within the bounded cellular volume and off-resonance effects give rise to experimentally significant systematic artifacts in microscopic images because the timescale for data collection can be comparable to the time needed for a mobile water molecule to diffuse across a cell. We are developing a quantitative theory to account for the new physical phenomena that affect MRI microscopy, and are modeling these phenomena. We have been implementing numerical simulation packages including: - Qualitative predictions of the effects of the variation of physical parameters on experimental images based on dimensional analysis, scaling, and intermediate asymptotics useful for guiding the design of experiments; - Efficient simulation packages for one and two dimensional problems based on rigorous, quantitative theory including the effects of saturating resonant radiation fields, systematic off-resonance effects due to the presence of strong magnetic field gradients, diffusive transport in the presence of boundaries; - New numerical techniques for predicting the effect of arbitrary resonant fields on spin systems and for solving the so-called steady-state saturation problem which has perplexed a number of earlier workers. This theoretical and computational work has been very helpful in guiding the development of experimental protocols.